# Random Walk in Two-Dimension

## The random walk is an important model for stochastic processes with applications in physics, statistics, finance, etc.

The random walk is an example of a **stochastic** or **random process**. Here, the outcome at time *t+1* is independent of the outcome at time* t*. Random walks have applications in several disciplines such as mathematics, physics, biology, chemistry, finance, etc. For example, a random walk can be used in finance to model stock prices, or in physics to model Brownian motion.

In this article, we present an example of a discrete random walk in two-dimension.

**Problem definition**: Suppose we start at the origin on a two-dimensional (2D) lattice on the *x-y* plane, what is the location of the random walk after *n* steps?

## Pseudocode for the random walk in 2D

Start at the origin (x = 0, y = 0)Toss a hypothetical die with 4 faces (1, 2, 3, 4)If the die returns 1, take arightstepIf the die returns 2, take aleftstepIf the die returns 3, take a stepupIf the die returns 4, take a stepdownRepeat the process for all n stepsGenerate a visualization to show trajectory of random walk

## Python Implementation

import numpyimport pylabimport randomn = 100000x = numpy.zeros(n)y = numpy.zeros(n)for i in range(1, n): val = random.randint(1, 4) if val == 1: x[i] = x[i - 1] + 1 y[i] = y[i - 1] elif val == 2: x[i] = x[i - 1] - 1 y[i] = y[i - 1] elif val == 3: x[i] = x[i - 1] y[i] = y[i - 1] + 1 else:

x[i] = x[i - 1] y[i] = y[i - 1] - 1

## Generate Visualization

pylab.title("Random Walk ($n = " + str(n) + "$ steps)")pylab.plot(x, y)pylab.xlabel("x", size=12)pylab.ylabel("y", size=12)pylab.savefig("rand_walk"+str(n)+".png",bbox_inches="tight",dpi=600)pylab.show()

## Output Example 1

## Output Example 2

We observe that the trajectory is random and different each time we run the code.

In summary, we’ve introduced a very simple example of a discrete random walk in 2D. Random walks are very important in probability theory and have applications in mathematics, physics, biology, chemistry, finance, etc.